# Ideal gases

> CIE A-Level Physics · CIE 9702
> Source: https://www.owlsprep.com/study/cie-9702-u19-overview/
> Weight: n/a

This unit explores the macroscopic behaviour and microscopic kinetic model of ideal gases, connecting measurable properties like pressure, volume and temperature to molecular motion, a core foundation for A-level thermodynamics.

**Prerequisites:** Basic thermal physics concepts (temperature, pressure); Newton's laws of motion and energy concepts

## Learning objectives

- State and apply the empirical gas laws to describe the behaviour of ideal gases
- Derive and use the ideal gas equation of state for solving gas-related problems
- State and explain the core assumptions of the kinetic theory of gases model
- Relate macroscopic gas properties to microscopic molecular motion
- Calculate root mean square speed and internal energy for an ideal gas

## Unit at a glance

This unit follows a logical arc from empirical observation to microscopic explanation: we start with experimentally derived gas laws that describe how ideal gases respond to changes in pressure, volume and temperature. These separate laws are combined into the unified ideal gas equation of state, the core macroscopic tool for ideal gas calculations.

We then move from bulk observations to a microscopic explanation of gas behaviour using the kinetic theory of gases, which connects bulk properties to the random motion of individual molecules. The unit concludes by applying this model to calculate root mean square molecular speed and define the internal energy of an ideal gas.

This unit is split into 5 connected sub-topics, ordered by increasing conceptual complexity:
- [Gas laws](https://www.owlsprep.com/study/cie-9702-u19-gas-laws/) — Learn the three empirical gas laws relating pressure, volume and absolute temperature.
- [Ideal gas equation of state](https://www.owlsprep.com/study/cie-9702-u19-ideal-gas-equation-of-state/) — Combine the gas laws into the unified ideal gas equation and apply it to problems.
- [Kinetic theory of gases](https://www.owlsprep.com/study/cie-9702-u19-kinetic-theory-of-gases/) — Explore core assumptions of kinetic theory and derive the pressure-motion relationship.
- [Root mean square speed](https://www.owlsprep.com/study/cie-9702-u19-root-mean-square-speed/) — Calculate and interpret root mean square speed for ideal gas molecules.
- [Internal energy of ideal gas](https://www.owlsprep.com/study/cie-9702-u19-internal-energy-of-ideal-gas/) — Understand why internal energy of an ideal gas depends only on temperature.

## Common pitfalls

- **Wrong:** Using Celsius temperature instead of absolute (Kelvin) temperature in gas calculations.
  - Why it fails: All ideal gas equations require temperature on an absolute scale; using Celsius gives systematically incorrect results.
  - Correct: Always convert temperature from °C to Kelvin by adding 273.15 before starting calculations.
- **Wrong:** Confusing the molar gas constant $R$ and Boltzmann constant $k_B$ in the ideal gas equation.
  - Why it fails: Mixing up these constants leads to incorrect units and values, depending on whether you use moles or number of molecules.
  - Correct: Use $R$ with moles $n$ ($PV = nRT$) and $k_B$ with number of molecules $N$ ($PV = Nk_B T$).
- **Wrong:** Assuming real gases follow the ideal gas law exactly at all conditions.
  - Why it fails: The ideal gas model relies on simplifying assumptions that break down at high pressure or low temperature.
  - Correct: Use the ideal gas law as an approximation, which is most accurate for low pressures and high temperatures.

## Cheatsheet

| Key Concept | Formula |
| --- | --- |
| Ideal gas equation (moles) | $PV = nRT$ |
| Ideal gas equation (molecules) | $PV = Nk_B T$ |
| Kinetic theory pressure equation | $PV = \frac{1}{3}Nm<c^2>$ |
| Root mean square speed | $v_{rms} = \sqrt{<c^2>}$ |
| Average kinetic energy per molecule | $\frac{1}{2}m<c^2> = \frac{3}{2}k_B T$ |
| Internal energy (monatomic ideal gas, n moles) | $U = \frac{3}{2}nRT$ |

## What's next

Begin your study of this unit with the first sub-topic on empirical gas laws, which forms the foundation for all further concepts about ideal gases. Work through each sub-topic in order, as every topic builds directly on knowledge from the previous one. Once you complete all sub-topics in this unit, you can progress to the next unit covering core thermodynamics concepts.

- [Gas laws](https://www.owlsprep.com/study/cie-9702-u19-gas-laws/)
- [Ideal gas equation of state](https://www.owlsprep.com/study/cie-9702-u19-ideal-gas-equation-of-state/)
- [Kinetic theory of gases](https://www.owlsprep.com/study/cie-9702-u19-kinetic-theory-of-gases/)

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